To prove the PBW theorem, Davison–Meinhardt first formulate a version of (REF ) for the analogous algebra of constructible sheaves on the coarse spaces \(X(d)\) . The map \(\pi :\mathcal {X}(d)\rightarrow X(d)\)  can be approximated by proper maps and thus \(\pi _*\)  commutes with vanishing cycles. Therefore the sheaf version of (REF ) for a general potential follows formally from the one for \(W=0\) . The sheaf version of (REF ) for \(W=0\)  follows from an explicit description of the summands of \(R\pi _*IC_{\mathcal {X}(d)}\)  due to
Meinhardt–Reineke [1]}.
